chemistry is life

[logistic_guy]

If you put $\displaystyle 120$ volts of electricity through a pickle, the pickle will smoke and start glowing orange-yellow. The light is emitted because $\displaystyle \text{sodium}$ ions in the pickle become excited; their return to the ground state results in light emission. $\displaystyle \bold{(a)}$ The wavelength of this emitted light is $\displaystyle 589 \ \text{nm}$. Calculate its frequency. $\displaystyle \bold{(b)}$ What is the energy of $\displaystyle 1.00 \ \text{mol}$ of these photons (a mole of photons is called an Einstein)? $\displaystyle \bold{(c)}$ Calculate the energy gap between the excited and ground states for the $\displaystyle \text{sodium}$ ion. $\displaystyle \bold{(d)}$ If you soaked the pickle for a long time in a different salt solution, such as $\displaystyle \text{strontium chloride}$, would you still observe $\displaystyle 589$-$\displaystyle \text{nm}$ light emission?

 

[logistic_guy]

$\displaystyle \bold{(a)}$

I am sure that you noticed that the wavelength is very very tiny, so it must depend on the speed of light.

The relation between the speed of light, frequency, and wavelength is:

$\displaystyle c = f\lambda$

Then, the frequency of the emitted light is:

$\displaystyle f = \frac{c}{\lambda} = \frac{3 \times 10^8}{589 \times 10^{-9}} = 5.09 \times 10^{14} \ \text{Hz}$

 

[logistic_guy]

$\displaystyle \bold{(b)}$

This is the beauty of chemistry (and physics). Once we have the frequency in our hands, we can calculate the energy of one photon.

$\displaystyle E = fh$

where $\displaystyle h$ is the $\displaystyle \text{Planck}$'s constant.

Then,

$\displaystyle E =5.09 \times 10^{14}(6.626 \times 10^{-34}) = 3.37 \times 10^{-19} \ \text{J}$

 

[logistic_guy]

$\displaystyle \bold{(b)}$

This is the beauty of chemistry (and physics). Once we have the frequency in our hands, we can calculate the energy of one photon.

$\displaystyle E = fh$

where $\displaystyle h$ is the $\displaystyle \text{Planck}$'s constant.

Then,

$\displaystyle E =5.09 \times 10^{14}(6.626 \times 10^{-34}) = 3.37 \times 10^{-19} \ \text{J}$

To get $\displaystyle 1 \ \text{mol}$ of these photons, we just need to multiply the energy by $\displaystyle \text{Avogadro}$ number.

Then,

$\displaystyle E_{1 \ \text{mol}} = E N_A = 3.37 \times 10^{-19}(6.022 \times 10^{23}) = 203 \ \text{kJ/mol}$

 

[logistic_guy]

$\displaystyle \bold{(c)}$

The energy gap $\displaystyle \Delta E$ is the same as the energy $\displaystyle E$ of one photon.

Therefore,

$\displaystyle \Delta E = E =3.37 \times 10^{-19} \ \text{J}$

 

[logistic_guy]

$\displaystyle \bold{(d)}$

Of course, you would not!

Why?

Simply because different elements emit different wavelengths of light. In other words, the energy gap is unique!